午夜福利1000集合

#11 Club shuffling

Set by Peter Hajek and Sam Hartburn

I play golf with three of my friends, and we split into two teams for each game, one team against the other. We change partners after each game until every possible pair of teams has played.

How many games will be played altogether?

If I win all my games, how many games can each other person win? What if I lose all my games?

If we add two more friends to the group and play in two teams of three instead, how does this change the answers to the questions above?

Solution next week

#10 Chairs in pairs

Solution

There are n seats in which to place Abbie. For each, there are n-1 places to seat Bryn. So there are n(n-1) ways to seat them both. Considering adjacent ways, there are n-1 ways to do this among n seats. For each, there are two arrangements (AB and BA), hence there are 2(n-1) ways they can be adjacent. So the probability of this is 2(n-1)/n(n-1) = 2/n. For n=4, this is 1/2. For n=20, this is 1/10. When seating people in pairs, we can think of them as one person taking up two chairs. We therefore want to seat two pairs, AA and BB, plus 18 singles, so imagine you arrange AA and BB among 20 positions, as in the second part of this BrainTwister, then swap their chairs for double chairs to seat their partners. The probability is therefore also 1/10.